Normalized defining polynomial
\( x^{21} + 21 x^{19} + 189 x^{17} + 952 x^{15} - 16 x^{14} + 2940 x^{13} - 224 x^{12} + 5733 x^{11} - 1232 x^{10} + 7007 x^{9} - 3360 x^{8} + 5228 x^{7} - 4704 x^{6} + 2639 x^{5} - 3136 x^{4} + 1505 x^{3} - 784 x^{2} + 581 x - 139 \)
Invariants
| Degree: | $21$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[3, 9]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(-2246296731094095348200059697976870763=-\,7^{35}\cdot 181^{3}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $53.83$ | magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
| |
| Ramified primes: | $7, 181$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{13} a^{18} + \frac{5}{13} a^{17} + \frac{5}{13} a^{16} - \frac{6}{13} a^{15} + \frac{5}{13} a^{14} - \frac{3}{13} a^{13} + \frac{2}{13} a^{11} - \frac{3}{13} a^{10} + \frac{4}{13} a^{9} - \frac{3}{13} a^{8} + \frac{4}{13} a^{7} - \frac{6}{13} a^{6} + \frac{6}{13} a^{5} - \frac{1}{13} a^{4} - \frac{6}{13} a^{3} + \frac{2}{13} a^{2} + \frac{1}{13} a - \frac{6}{13}$, $\frac{1}{13} a^{19} + \frac{6}{13} a^{17} - \frac{5}{13} a^{16} - \frac{4}{13} a^{15} - \frac{2}{13} a^{14} + \frac{2}{13} a^{13} + \frac{2}{13} a^{12} + \frac{6}{13} a^{10} + \frac{3}{13} a^{9} + \frac{6}{13} a^{8} - \frac{3}{13} a^{6} - \frac{5}{13} a^{5} - \frac{1}{13} a^{4} + \frac{6}{13} a^{3} + \frac{4}{13} a^{2} + \frac{2}{13} a + \frac{4}{13}$, $\frac{1}{13} a^{20} + \frac{4}{13} a^{17} + \frac{5}{13} a^{16} - \frac{5}{13} a^{15} - \frac{2}{13} a^{14} - \frac{6}{13} a^{13} - \frac{6}{13} a^{11} - \frac{5}{13} a^{10} - \frac{5}{13} a^{9} + \frac{5}{13} a^{8} - \frac{1}{13} a^{7} + \frac{5}{13} a^{6} + \frac{2}{13} a^{5} - \frac{1}{13} a^{4} + \frac{1}{13} a^{3} + \frac{3}{13} a^{2} - \frac{2}{13} a - \frac{3}{13}$
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
| Rank: | $11$ | magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
| |
| Torsion generator: | \( -1 \) (order $2$) | magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
| |
| Fundamental units: | Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH) | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 14319829769.0 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A solvable group of order 8232 |
| The 55 conjugacy class representatives for t21n45 are not computed |
| Character table for t21n45 is not computed |
Intermediate fields
| \(\Q(\zeta_{7})^+\) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Frobenius cycle types
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | 53 | 59 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/LocalNumberField/2.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/2.3.0.1}{3} }$ | ${\href{/LocalNumberField/3.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }$ | ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }$ | R | $21$ | ${\href{/LocalNumberField/13.7.0.1}{7} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{8}$ | $21$ | ${\href{/LocalNumberField/19.3.0.1}{3} }^{7}$ | ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }$ | ${\href{/LocalNumberField/29.7.0.1}{7} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/31.3.0.1}{3} }^{7}$ | $21$ | ${\href{/LocalNumberField/41.7.0.1}{7} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{14}$ | ${\href{/LocalNumberField/43.7.0.1}{7} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ | $21$ | ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }$ | ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }$ |
In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.
Local algebras for ramified primes
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
| 7 | Data not computed | ||||||
| $181$ | $\Q_{181}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 181.2.1.2 | $x^{2} + 362$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 181.2.1.2 | $x^{2} + 362$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 181.2.1.2 | $x^{2} + 362$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 181.7.0.1 | $x^{7} - x + 2$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | |
| 181.7.0.1 | $x^{7} - x + 2$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | |